Of course Eq. (REF ) and \((\ref {bound1})\)  reduce to Eq. (REF ) and (REF ) when \(\tilde{p}(\sigma ,\phi ) = p(\sigma ,\phi ) \) . Thus the bound is saturated by the minimal distribution of Ref. [1]}. We emphasize that the bound in Eq. (REF ) is tighter than the hysteretic thermodynamic uncertainty relation derived in Ref. [2]}, [3]}.
